For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...
9
votes
1answer
512 views
How to understand spectral decomposition geometrically
Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have
$$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$
and
$$A^{-1} = ...
9
votes
5answers
2k views
Relation between Cholesky and SVD
When we have a symmetric matrix $A = LL^*$, we can obtain L using Cholesky decomposition of $A$ ($L^*$ is $L$ transposed).
Can anyone tell me how we can get this same $L$ using SVD or Eigen ...
9
votes
3answers
2k views
Physical Meaning of Null Space of a Matrix
What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions... my textbook gives me the definition, but I just don't "get" it.
E.g.: I ...
9
votes
2answers
1k views
Calculating RGB plus Amber
I'm currently working on a wide gamut light source using red, green and blue LED emitters. From an internal xyY (or CIE XYZ) representation, I can reach any color or color temperature via a 3x3 ...
9
votes
1answer
291 views
How many $n\times m$ binary matrices are there, up to row and column permutations?
I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations.
If $\sim$ is the equivalence relation on $n\times m$ binary matrices such ...
9
votes
3answers
195 views
What are mandatory conditions for a family of matrices to commute?
Suppose that there are some matrices. Each matrix in the set must commute with another in the set.
What are the mandatory conditions for this?
9
votes
1answer
141 views
Matrix algorithm convergence
Suppose I start with a $n \times n$ matrix of zeros and ones:
$$
\begin{bmatrix}
0 & 0 & 0 & 1 & 1\\
1 & 1 & 1 & 1 & 1\\
1 & 1 & 1 & 1 & 1\\
1 ...
9
votes
2answers
221 views
What can we say about $(I-AD)^{-1}$ if $D$ is a diagonal matrix?
Assume we know that square matrices $A$ and $(I-AD)^{-1}$ are invertible and also $D$ is a diagonal matrix. Also assume that $A$ is a symmetric matrix. My question is when we can express ...
9
votes
5answers
291 views
Smallest Non-negative number in a matrix
There is a question I encountered which said to fill an $N \times N$ matrix such that each entry in the matrix is the smallest non-negative number which does not appear
either above the entry or to ...
9
votes
1answer
533 views
Does equality of characteristic polynomials guarantee equivalence of matrices?
I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and ...
9
votes
1answer
455 views
Do these matrix rings have non-zero elements that are neither units nor zero divisors?
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$.
In addition, suppose that $R$ is a ring in which every non-zero element is ...
9
votes
2answers
524 views
Augmented Reality Transformation Matrix Optimization
i am a software developer, i'm working on an Augmented Reality system.
I'd like to receive some advice in order to optimize my math model. My program has to be slim and fast.
Here's the situation:
...
9
votes
4answers
1k views
Square root of a matrix
Under what conditions does a matrix $A$ have a square root? I saw somewhere that this is true for Hermitian positive definite matrices(whose definition I just looked up).
Moreover, is it possible ...
9
votes
1answer
316 views
Detecting symmetric matrices of the form (low-rank + diagonal matrix)
Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
8
votes
3answers
268 views
$A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$
I am stuck on this simple question for a long time.
If $A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$?
I tried finding a counter-example as well as tried ...
8
votes
4answers
1k views
Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]
Possible Duplicate:
Units and Nilpotents
Hi all wondering if I could get a bit of help with this, given $A^{2012}=0$ prove $(A+I)$ is invertible and find an expression for $(A+I)^{-1}$ in ...
8
votes
4answers
493 views
“weird” ring with 4 elements - how does it arise?
For no real reason, I did a classification of (associative, with a 1) rings with less than 8 elements (they all happen to be commutative). Most of the rings I got were of a type I knew - namely: ...
8
votes
3answers
249 views
Is $\;\det(A^n) =\left(\det (A)\right)^n\;$?
How can the value of $\;\det\left(A^{11}\right)\;$ be calculated from $\;\det(A)$?
Generally how can $\;\det\left(A^n\right)\;$ be obtained from $\;\det(A)$?
8
votes
7answers
266 views
Is the square root of a triangular matrix necessarily triangular?
$X^2 = L$, with $L$ lower triangular, but $X$ is not lower triangular. Is it possible?
I know that a lower triangular matrix $L$ (not a diagonal matrix for this question),
$$L_{nm} \cases{=0 & ...
8
votes
1answer
961 views
Is the matrix $A$ diagonalizable if $A^2=I$
If $A$ is an involutory matrix, i.e. $A^2=I$, then is $A$ diagonalizable?
8
votes
5answers
170 views
Matrices with eigenvalues 0 and 1
How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1?
My attempt:
I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
8
votes
3answers
293 views
Why is the determinant of a symplectic matrix 1?
suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix}
0 & E_n\\
-E_n&0
\end{pmatrix}$$
where $E_n$ represents identity matrix.
if $A$ satisfies $$AJA^T=J$$
How to figure out ...
8
votes
3answers
222 views
How to show path-connectedness
Well, I am not getting any hint how to show $GL_n(\mathbb{C})$ is path connected. So far I have thought that let $A$ be any invertible complex matrix and $I$ be the idenity matrix, I was trying to ...
8
votes
4answers
2k views
How do I calculate the $p$-norm of a matrix?
I know that the $p$-norm for a matrix is:
$$\|A\| = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}$$
but I don't know what this really means.
So how would I compute the $2$-norm, $3$-norm, etc for the ...
8
votes
3answers
296 views
$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.
The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful.
Let $A,B$ be $m \times n$ and $n ...
8
votes
2answers
110 views
Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?
Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
8
votes
3answers
164 views
Minimize $||Ax-b||$ but for $A$, not $x$
I have a machine learning regression problem. I need to minimize
$$
\sum_i||Ax_i-b_i||_2^2
$$
However I am trying to find matrix $A$, not the usual $x$, and I have lots of example data for $x_i$ and ...
8
votes
1answer
206 views
The first column of the $n$th power for a triangular matrix
I have found a interesting thing but I cannot prove it. Given $k_i$ are positive for any $i\geq1$, and we have $M+1$ by $M+1$ matrix $A$, which is
$$
A=\left[\begin{array}{ccccc}
0\\
k_{1} & 0\\
...
8
votes
2answers
313 views
Significance of the following Matrix?
I am unfamiliar with advanced Matrix theory (nor am I a mathematician), so please bear with me.
Is there anything significant about the following Matrix structure? Are there any special symmetries or ...
8
votes
1answer
2k views
Is it faster to multiply a matrix by its transpose than ordinary matrix multiplication?
I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
8
votes
1answer
123 views
Is there a name referring to this result?
For any real $m \times n$ matrix $A$, it seems that
$$\det(I_n + A^{T}A) = \det(I_m + AA^{T}) $$
always holds, where $I_n$ is the identity matrix of size $n$.
Though I have not tried to prove this ...
8
votes
2answers
423 views
Span of Permutation Matrices
The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, ...
8
votes
2answers
211 views
Probability of determinants being coprime
I have a question that is not of particular significance, but I would love to understand the underlying principles.
Suppose we have two square 3x3 matrices, $M_1$ and $M_2$ with
$$M_1 = ...
8
votes
1answer
745 views
Probability that a random binary matrix is invertible?
What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible?
Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$.
Is there an explicit formula as a ...
8
votes
1answer
70 views
Find $M$, where $M^7=I$ and $M\neq I$, $M$ has only 0's and 1's.
Find a $3 \times 3 $ matrix $M$ with entries 0 and 1 only such that $M^7=I$ and $M\neq I$.
This was a short question in a recent exam. I tried with permutation matrices but couldn't find $M^{odd}=I$ ...
8
votes
3answers
163 views
Find a ternary $4\times 39$ matrix satisfying the conditions below
Can you find a matrix $A_{4\times39}$ with elements from $\{-1,0,1\}$ so that
No column is all zero.
All columns are different.
No column is $-1$ times another column.
Each row consists of $13$ of ...
8
votes
1answer
171 views
How prove this linear algebra $AB=BA$?
Suppose $A,B\in M_{n}(\Bbb C)$ satisfies for $\forall a,b\in \Bbb C,aA+bB$ is always diagonalizable.
Show that
$$AB=BA.$$
8
votes
1answer
208 views
Dimensionality of null space when Trace is Zero
This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216). I've succeeded in proving the first three parts, but the most ...
8
votes
1answer
719 views
In-place inversion of large matrices
In Solving very large matrices in "pieces" there is a way shown to solve matrix inversion in pieces. Is it possible to apply the method in-place?
I am refering to the answer in the ...
8
votes
1answer
793 views
Upper-triangular matrix is invertible iff its diagonal is invertible: C*-algebra case
Exercise 1.14 of the book Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove, that upper triangular matrix with elements from some C*-algebra $A$ is invertible in ...
8
votes
3answers
405 views
Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?
It's easy for scalars, $(\exp(a(x)))' = a' e^a$. But can anything be said about matrices? Do $A(x)$ and $A'(x)$ commute such that $(\exp(A(x)))' = A' e^A = e^A A'$ or is this only a special case?
8
votes
1answer
104 views
The Determinant of a Sum of Matrices
Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$,
\begin{align}
\det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma},
\end{align}
...
8
votes
1answer
100 views
Möbius function from random number sequence
Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}.
Then plug it ...
8
votes
2answers
122 views
Why is there no simpler form of a matrix than the Jordan or Frobenius normal form?
The Jordan and Frobenius normal forms of a linear map $A:\Bbb R^n \rightarrow \Bbb R^n$ seem to be maximally simple representations of $A$ in the sense that one of them contains as few nonzero entries ...
8
votes
2answers
122 views
Solving matrix equations of the form $X = AXA^T + C$
I'm trying to solve this matrix equation:
$$X = AXA^T + C$$
In particular,
$$
X = \begin{bmatrix} 1.5 & 1 \\ -0.7 & 0 \end{bmatrix}
X
\begin{bmatrix} 1.5 & -0.7 \\ 1 & 0 ...
8
votes
1answer
257 views
How to diagonalize this matrix?
Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads
$$
M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0)
$$
where the $a_i$'s are given positive natural numbers.
...
8
votes
1answer
218 views
Matrix factorization
I'd like to factorize matrices as follows:
$$ \left(\begin{array}{cc}X_1&X_2\\X_3&X_4\end{array}\right) = ...
8
votes
0answers
192 views
Limit of sequence of growing matrices
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
...
8
votes
0answers
177 views
Inverse of Toeplitz Matrix Property
Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 ...
8
votes
0answers
157 views
$\mathcal{O}(n,\mathbb R)$ spans $\mathcal{M}(n,\mathbb R)$
Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb ...
